Improved probabilistic method for power system dynamic stability studies

K.W.Wang, C.Y.Chung, C.T.Tse and K.M.Tsang

Abstract: An improved probabilistic method using moments and cumulants of random variables for power system dynamic stability studies is presented. The uncertainties considered are system multi- operating conditions derived from operating curves of load powers and generations. The generator state and nodal voltages are solved from a stochastic load flow calculation. By means of the first and second order eigenvalue sensitivity representation, moments and cumulants of eigenvalues are determined from the statistical characteristics of nodal voltages and nodal injections. Probabilistic densities and conditional probabilities of critical eigenvalues are calculated from the Gramxharlier series. In this method, random variables can have an arbitrary distribution. Dependencies among random . variables and the interaction between expectation and covariance are all considered. Examination on two test systems shows that the proposed method can relievehprove the conflict between computing requirement and precision.

1 Introduction

In power system operation there are many fluctuations and random factors such as the variations of load powers and generations, changes in network configuration and system parameters, as well as the measure and forecast errors. It is expedient to consider more system operating conditions and random factors in one numerical calculation.

The probabilistic analysis for power system dynamics is to determine the probabilistic distributions of critical eigen- values and the conditional probability of overall system dynamic instability. [l] gave the first attempt in this aspect in whch the uncertainties are the measure and forecast errors of nodal powers and system parameters. All random variables are assumed subject to a normal distribution. Probabilistic densities of real parts of eigenvalues are deter- mined from the expectations and covariances of some sys- tem random parameters with a linearised system model in state space form. The stability probability of the whole sys- tem is then calculated on a two-machine system from the concept of a joint normal distribution. Ths method was also developed to contain random variables with any type of distribution by using a moment approach [2]. The con- cept of stochastic stability was also employed to determine the statistical property of dynamic stability limit curves on a single-machine infmite-busbar system [3]. [4] described a probabilistic dynamic model of an induction machine, in which the equivalent system matrix A , was determined from eigenvalues and eigenvectors by a heuristic approach. To consider multi-operating conditions of a power system, higher order cumulants and a Gram-Charlier series were used in [5], from which the statistical property of a simplis- tic stability criterion dPJdS, > 0 is computed.

0 IEE, 2000 IEE Proceedings online no. 20000025 DOL lO.l049/ipgtd:20000025 Paper first received 17th February and in revised form 22nd July 1999 The athors are with the Department of Electrical Engineering, Hong Kong Polytechnic University, Hong Kong

Up to the present, the main difficulties in probabilistic dynamic stability studies are still the computing precision and computation requirement (i.e. CPU time and mem- ory). A normal distribution represented by expectation and covariance is perhaps the simplest one [l, 31. The moment method [2] allows the accurate representation of dependen- cies among random variables, but requires a great amount of computation for the calculation of hgher order eigen- value moments. Under the assumption of statistical inde- pendence, various order cumulants of eigenvalues can be computed very quickly [5]. The nonlinear relationship from eigenvalues to nodal voltages or nodal injections is another factor that increases the computation requirement.

To relieve the conflict between computation requirement and computing precision, a hybrid algorithm composed of the central moment and the cumulant is introduced in this paper. Different degrees of approximation are applied for the calculation of various order moments and cumulants according to their effects on the eigenvalue distribution. Since the effect of multi-operating conditions on dynamics is considered under the foundation of operating curves of nodal injections, a direct comparison is conveniently given between the solution obtained from the proposed method and that from a deterministic approach. Moreover, with the system state space equation constructed from a plug-in modelhg technique (PMT) [6], analytical formulae of the first and second eigenvalue sensitivities for a multimachine system are systematically derived regardless of the system complexity.

2 eigenvalues

The variation of the system operating state due to nodal power changes and other factors can be regarded as a sto- chastic process. All system variables contained in or affected by the process are considered as random variables, and so are the eigenvalues. The statistical nature of a ran- dom variable can be described by its numerical characteris- tics such as the central moment or cumulant.

Cumulant simulation for the statistical nature of

IEE Proc-Gener. Trunsm. Distrib., Vol. 147, No. I , January 2000 37

When the cumulant method is applied for probabilistic eigenvalue analysis [5], the cumulants of original random variables, such as nodal injections in this paper, are first determined. Then, various order eigenvalue cumulants are calculated from the cumulants of nodal injections with the assumption of statistical independence. The distribution curves or conditional probabilities of eigenvalues are esti- mated from eigenvalue cumulants by a series representa- tion, such as the Gran-Charlier series [8].

However, to inspect the dlstribution of eigenvalues and to determine the necessary order number of the cumulant, the cumulants of a critical eigenvalue of a three-machine system (to be discussed in Section 7) are obtained from its 480 sample values. The 480 samples of the critical eigen- value are calculated from 480 sets of sample data of nodal injections by repeated deterministic eigenvalue calculation. The probabilistic density of the real part of the critical eigenvalue is obtained as curve 1 in Fig. 1. Probabilistic density curves are also obtained by cumulant simulation using the first two, three, five and seven order cumulants, respectively. Curve 2 in Fig. 1 is the normal simulation determined only from expectation and variance (first two order cumulants). Curve 3 is the simulation from a Gram- Charlier series with the first five order cumulants used. A simulation curve using the first three order cumulants is located between curves 2 and 3, whde a curve using the fust seven order cumulants is very close to curve 3, so simula- tions using the first three and the frst seven order cumu- lants are not given in Fig. 1. Because the right 'tail' of the density function curve is of more interest in conditional probability, it is obvious that curve 3 is more desirable. Therefore, the first five order cumulants will be used in this paper in whch the second and third order cumulants are determined by the moment method [2], to consider the sta- tistical dependencies among variables.

machine1

g 5.5 P 4.5 ._ - ; 3.5 .- U)

U E 2.5

1.5 0.5

- machine i -

-.- ! 3 ; 4 ; * 7 0 0 6 0 2 8 - J

real part of eigenvalue Fig. 1 (i) actual distribution (ii) normal simulation (iii) simulation by first five order cumulants

D i s t r h t b n ofthe r e d part o f a critical eigenvalue

A A

AIq

Furthermore, numerical calculation shows that the hgher the order of cumulant in a Gram-Charlier series, the weaker its effect will be on the entire distribution. To reduce the amount of computation, different degrees of approximation should be adopted for computing different order moments or cumulants of eigenvalues. The consider- ation of variable dependence in central moments and the simplicity of the calculation of higher order cumulants form a hybrid probabilistic algorithm for power system dynamic stability studies.

A A

AIyi AVq AVyi

3 Multimachine system representation

In order to determine the relationshp between eigenvalues and nodal voltages, a versatile plug-in modelling technique

38

v v

(PMT) [6] is used to form the system state space equalion, from which the generators and control systems can be rep- resented to any desired complicated degree. When the load powers are represented as

PL = PLOV" Q L = Q L o V ~ (1) the linearised representation of d V / d will yield four conipo- nents [7] for each nodal load, which can be 'merged' into a nodal admittance matrix of the entire power network. With a linearised coordinate transformation included in the machine model, all generators are directly connected with an external power network described by a linearised nodal voltage equation, as shown in Fig. 2. (In Fig. 2, subscripts 'x' and 'y' indicate a rectangular network frame).

t t

Fig. 2 Multinachine system representation

In the PMT technique, the state space equation of the entire system is formatted in the Appendix (Section I 1. 1), while the initial state of generators is obtained by a proba- bilistic load flow calculation in Section 11.2.

4 Numerical characteristics of eigenvalues

4. I Expectation and variance Expectations and covariances of eigenvalues are computed from the expectations and covariances of nodal voltages. A particular complex eigenvalue dk can be analytically expressed as a nonlinear function of the nodal voltage vec- tor Vas

x k = Gk(V) (2) In a power system of N nodes, the voltage vector contains 2N real components as V = [VI, V,, ..., V2,,,lT. Eqn. z! can be expanded in a Taylor series, and the approximate r1:pre- sentation with second order terms retained is:

-

The expegtion operator is expressed as (.I or E(.). Con- sidering A 6 = 0 (for i = I, 2, ... 2N) and the nodal voltage covariance A V A V = CFJ between V, and 5, the expecta- tion of eigenvalue dk is gven as I

, (4) where, X k o = G k ( V ) is solved from system matrix Ih in eqn. 20. Hk is the second order term given by

2 N 2 N I I

From eqns. 4 and 5, the effect of covariance has been considered in the calculation of eigenvalue expectation. To determine eigenvalue covariances, eqn. 2 is linearised as

IEE Proc.-Gener. Transm. Dislrib., Vol. 147, No. 1, Januar.v 2000

eqn. 6 and expressed in terms of the first order derivative J&

2 N

A k = X k O + (JAl;,*A&) (6) i=l

The covariance between eigenvalues A, and & is

Substituting eqns. 4 and 6 into eqn. 7 gives 2 N 2 N

2=13=1

Since v and C y have been obtained from the probabilistic load flow calculation, the first two order moments of eigen- values can be calculated from eqns. 3 and 8. The diagonal elements of matrix C, are the variances of the eigenvalues, and the off-diagonal elements are the covariances between eigenvalues. By making the array of eigenvalues real (e.g. if AR,R+I = a * jp, then AR = a, &+I = p), CL will also be a real matrix. Even for the full eigenvalue calculation, only the diagonal 2 x 2 blocks of CA need to be computed.

4.2 Third order central moment The linearised relationship between eigenvalue and nodal injection power is obtained from eqns. 6 and 24, and can be written in the following matrix form

AA = J x A V = J x J G I A S = J A S (9) where S stands for the nodal injection vector, and J = J, J y ' . Considering the dependence among nodal powers, the third order moment MA3) of a particular eigenvalue & is calculated from the thrd order central moments of nodal injections as

2 N 2 N 2 N

z=1 j=1 n=l

where J,d is the element of matrix J.

4.3 Higher order cumulants [8] Using eqn. 9 and omitting the dependence among input random variables, the fourth and ffth order cumulants of an eigenvalue i lk are approximately computed from the same order cumulants of nodal injections:

The computation of cumulants in eqn. 11 is very fast. However, referring to eqn. 10, if the fourth and fifth moments are applied [2] to replace eqn. 11, the computa- tion requirement will become much larger because of the dependence among nodal injections.

5 Probabilistic distribution of eigenvalues

The distribution of the real part of a particular eigenvalue & indicates the degree of stability for corresponding oscilla- tory mode. Suppose iLk = a k + jpk, each order cumulant of a k is the real part of the same order cumulant of Ak:

K&) = Re [KK)] (T = 2,3,4,5) (12)

Since the variance and the third order central moment of a random variable are just its second and third order cumu- lants, respectively, the probabilistic density of a k can be cal- culated from the Gram-Charlier series [SI represented in

IEE Proc -Gener Transm Distrib , Vol 147, No I , Junuary 2000

the Appendix (Section 11.3). The conditional probability can be expressed as

Qic

P { a < a,} = J' f ( a ) d a (13) -CO

where P{ .} stands for the conditional probability and Aa) is the probabihstic density function of a. If the limit varia- ble ac equals zero, eqn. 13 gives the probability of Ak locat- ing on the left half of a complex plane.

The conditional probability of a critical eigenvalue will approximately express the dynamic stability probability of the entire multimachine system.

The damping ratio Ek corresponding to a particular eigenvalue h k = ffk + jP, indicates the damping degree for the kth oscillatory mode and is defined as

[ k = - a k 1 dm (14)

According to the linearisation of eqn. 14, each order cumulant of is calculated from the cumulants of a k and &. The density function of 4 is then computed from Sec- tion 11.3, while the conditional probability is determined by eqn. 15, with the acceptable limit Ec equating to 0.1 [9] in this study:

p { c k > '$} = 1 - p { [ k < r c } = 1 - 1 f(WE -cc

(15) Details of eqns. 13 and 15 are shown as eqn. 27 in the

Appendix (Section 1 1.3).

6 Sensitivity formulae

Let w k and u k stand for the left and right eigenvectors cor- responding to mode k, respectively; the first and second order eigenvalue sensitivity representations are [lo]:

d W r d A +--uk + -~ avi avj

(17) In eqn. 17, the derivative of the left eigenvector W l is a lin- ear combination of all eigenvectors:

Therefore, only the first and second order derivatives of the system matrix A with respect to nodal voltages need to be determined (see the Appendix (Section 1 1.4)).

7 Casestudies

System I: In the 3-machine 9-bus system shown in Fig. 3, all machines are equipped with fast-acting static exciters and speed governors. Machine 3 is also equipped with a power system stabiliser. Block diagrams of all this control equipment are provided in [I 11. The normal operation

39

values of load powers and PV voltages are regarded as the corresponding expectations: SloadA = 125 + jSO(MVA), SIoa& = 100 + j35(MVA), s ,oudc = 90 + J~O(MVA), s ~ , = 113 +j6.7(MVA), P, = 85(MW), VG2 = 1.025@~), VG, = 1.04L0°(pu).

$ 2.0

.- F 1.0-

1.5-

- 0.5

&? Fig.3 3-machine system (system I)

-

-

time, hours a

2.5 r

0; ' 3 ' i ' + ' 9 ' 1'1 ' 1$' I b ' 1;' 1'9' dl ' 23' time, hours

b

1.01 gJ 1.00 B 0.99 p 0.98 0.97 0.961, , , , , , , , , , , , , , , , , , , , , , , 0'951 3 5 7 9 1 1 13 15 17 19 21 23

time, hours

C Fig. 4 ence voltuge

Dmly curye of (U) loadpowers, (b) generation powers and (e) r&r-

Load powers are transferred to equivalent admittances derived from eqn. 1. The standardised daily operating curves shown in Fig. 4 are used to create 480 power and voltage samples. Because the original curve of reactive power is relatively flat compared to the active power, to demonstrate the effectiveness of the present approach it is assumed to have ancharacteristic Q;,, = (1 + 2 x Bi,3/3 in the present study. P,, and Qi,t are values in standard oper- ating curves of active and reactive, respectively, for node i and the tth time interval.

From the created 480 power samples, the moments or cumulants of nodal injections can be determined. Numeri- cal characteristics of eigenvalues are computed and cor- rected from eqns. 4, 8, 10 and 11. Distribution densities and conditional probabdities of eigenvalues are calculated by eqns. 26 and 27. This system results in 27 eigenvalues composed of 5 pairs of complex eigenvalues (Table 1) and 17 real eigenvalues (Table 2). The fxst five order numerical characteristics of all real eigenvalues are listed in Tables 2 and 3. All real eigenvalues are stable due to their adeqyate probabilities of P{ A < 0}, while probabilities corresponding

40

to complex eigenvalues are listed in Table 1, and the 23rd and 25th eigenvalues are relatively critical.

Table 1: Complex eigenvalues of system I - -

Pia < 0) > 0.1) - No. iz P 5 11, 12 -2.1803 0.0721 0.9995 1.0000 1.OOOCI

15, 16 -1.1096 0.7881 0.8153 1.0000 1.OOOCl

19,20 -0.7532 0.7156 0.7250 1.0000 1.OOOCI

22,23 -0.4909 9.2558 0.0530 0.9920 0.0000

24,25 -0.3466 6.8038 0.0509 0.9969 0.0034.

Table 2: Cumulants of real eigenvalues of system I

No. MeanX Variance K i 3 ) Ki4) 1 -109.261 0.507E-01 0.544E-02 -0.130E-02

2 -105.621 0.502E-01 0.546E-02 -0.105E-02

3 -104.783 0.451E+00 0.176E+00 -0).431E+00

4 -74.429 0.893E+00 -0.482E+OO -0).167E+01

5 -73.236 0.105E+00 -0.165E-01 -0.464E-02

6 -67.618 0.132E+00 -0.232E-01 -0.882E-02

7 -35.241 0.234E-09 -0.168E-14 -0.467E-19

8 -16.665 0.409E-06 0.788E-10 -0.196E-12

9 -5.000 0.468E-09 -0.132E-15 -0.181E-18

10 -3.625 0.105E-03 0.919E-07 -0.166E-07

13 -1.999 0.235E-06 -0.868E-13 -0.131E-14

14 -1.455 0.159E-01 -0.189E-05 -0.307E-05

17 -1.000 0.614E-10 -0.107E-15 -0.566E-21

18 -0.999 0.379E-07 0.112E-11 -0.455E-14

21 -0.668 0.141E-06 -0.130E-10 -0.282E-14

26 -0.203 0.123E-10 0.573E-18 -0.225E-24

27 -0.017 0.204E-07 0.124E-12 -0.198E-17

k15) ~- -0.604E-43

-0.976E-03

-0.1 10E-01

0.599E-01

0.624E-02

0.655E-02

0.196E--23

-0.469E-15

0.587E-23

-0.174E-49

-0.660E-18

0.997E-07

0.850E-26

-0.250E-17

0.345E-17

-0.558E-30

-0.283E-21

Table 3: Cumulants of complex eigenvalues of system I

Real parts - Mean Variance KL3) KL4) K i 5 ) a Eigenvalues - (x 10-2) ( x 10-3) ( X 10-4) (X 1 0 9

11,12 -2.1803 2.4328 -0.0640 -0.4317 i.oa26 15,16 -1.1096 0.3487 0.0056 -0.0110 -0.0176

19,20 -0.7532 0.9526 0.0350 -0.1760 -0.2177

22,23 -0.4909 3.9392 3.6379 -12.4862 -74.06;47

-0.3466 1.5757 0.1491 -0.2040 -0.3245 - 24,25

Imaginary parts ~~

K (4) Kg(5) - Eigenvalues Mean Variance Kk3) P

11,12 0.0721 0.0384 -0.0003 -0.0003 0.0001

15,16 0.7881 0.0005 0.0000 0.0000 0.0000

19,20 0.7156 0.0012 0.0000 0.0000 0.0000

22,23 9.2558 2.9888 1.7011 -6.2820 -31.3655

6.8038 0.3007 0.0637 -0.0309 -0.0666 - 24,25

Damping ratios ~~

Mean Variance Kd3) Kd4) Kd5) Eigenvalues 5 ( x 10-2) ( x 10-3) (X 1 r 4 ) (x 10"')

11,12 0.9995 0.001 1 0.0000 0.0000 0.OOOOi

15,16 0.8153 0.2078 0.0001 0.0000 0.0000~ 19,20 0.7250 0.2381 -0.0017 -0.0078 0.0043

22,23 0.0530 0.0307 -0.0042 -0.0017 0.0010

24,25 0.0509 0.0325 -0.0004 -0.0001 0.0000

IEE Proc.-Gener. Trunsm. Distrib., Vol. 147, No. I , January 2000

For testing the validlty of the proposed algorithm, 480 sets of eigenvalues are solved by the deterministic approach from the 480 sample data of nodal powers. From these eigenvalue samples, the actual distribution of the real part of the 23rd eigenvalue is obtained as curve 1 in Fig. 5. Curve 2 is calculated from the proposed method. As explained in Section 2, Fig. 5 shows that the proposed method can give a more ‘useful’ eigenvalue distribution.

5 r I

c 4 - ‘li 5 1 i 3 -

r 7 - . - 0 ~ ~ 0 ~ 0 0 0 0 0 0 0 I , ,

real part of eigenvalue Fig. 5

~ actual distribution - simulation by proposed method

Distribution ofthe realpart of 23rd eigenvalue

Table 4: Comparison of expectations and standard devia- tions for critical eigenvalues

23 aa23 8 2 5 aa25

Linear model -0.5472 0.1903 -0.4507 0.0701 Proposed method -0.4909 0.1985 -0.3466 0.1255 Actual value -0.4880 0.2080 -0.3340 0.1747

Table 4 gives a comparison of the first two order moments of critical eigenvalues. The last row of Table 4 is calculated from the 480 sets of eigenvalue samples, while the first row is obtained by omitting the second terms of eqns. 4 and 8. From Table 4, the expectations solved by the proposed method are very close to their actual values due to the correction of eqn. 4. Certain errors still exist in eigenvalue variances due to only an approximate correction in eqn. 8.

-- - 7 24 6 1 -- 13 A A

-

G I 10 2- 8

L1 L2 L4 ---2

G6

I

I I 4 2 5 q 15&

0 0 G7 G4 G5

Fig.6 8-machine systenz (System II)

System ZL The 8-machine 25-bus system shown in Fig. 6 is modified from the 36-bus test system in the ‘Power sys- tem analysis software package (PSASP)’ by omitting DC h k s . All network parameters, nodal powers and control system parameters are obtained from PSASP. Dlfferent daily curves are assigned for nodal powers S,, to SL9, P G ] to P, and QG3, ea, respectively. The reference voltage at slack bus 24 varies according to the curve of Fig. 4c. The voltage dependent exponential constants in eqn. 1 are a =

IEE Proc.-Gener. Transm. Dislrib.. Vol. 147, No. I , January 2000

1.38 and b = 3.22 [6] for active and reactive powers, respec- tively. The solution gives 74 eigenvalues composed of 34 real eigenvalues and 20 pairs of complex eigenvalues. Seven pairs of complex eigenvalues have unsatisfied damping ratios, as listed in Table 5, in which three pairs are unsta- ble. The first five order cumulants of real parts of these eigenvalues are given in Table 6.

Table 5: Selected complex eigenvalues of system II

No.

45’46 52,53 58,59 60,61 62,63 64,65 73.74

h

-0.4283 -0.3694 -0.2865 -0.0353 -0.1382 -0.0954 -0.1978

- P 7.0792 14.5785 0.3531 5.8210 10.8349 7.8486 7.1538

- 5 0.0604 0.0253 0.6301 0.0061 0.0128 0.0122 0.0276

Ra < 01

1 .oooo 1 .oooo 1 .oooo 0.5606 1 .oooo 0.8908 0.8199

~~ ~

P I E > 0.11

0.0000 0.0000 0.8885 0.0066 0.0000 0.0000 0.0073

Table 6 Cumulants of real parts of selected complex eigen- values

Mean Variance Kk3) KL4) Kk5) h (x 10-9 (x (x 1w4) (x 1 0 9 No.

45,46 -0.4283 0.0815 0.0039 -0.0011 -0.0006 52,53 -0.3694 0.3214 0.0700 -0.0251 -0.0079 58,59 -0.2865 0.9075 -0.2913 -0.0252 0.0420 60,61 -0.0353 4.9278 0.0710 -1.9278 -6.6744 62.63 -0.1382 0.0287 -0.0021 0.0107 -0.0004

64,65 -0.0954 0.4392 0.1524 -0.2309 -0.2446 73,74 -0.1978 4.5689 -0.5763 -1.1106 2.3602

From Table 5, the 61st, 65th and 74th eigenvalues are critical because of their inadequate stability probabilities of P { a < O}; in particular, the 61st eigenvalue with stability probability 0.5606. Notice that the low probability value of

is due to its relatively larger variance of 0.049278 in Table 6, and the expectation of -0.0353 in Table 5 This means will vary over a relatively wide range with the load power variation.

The effect of higher order cumulants on the conditional probability can be examined by the second term in the right-hand side of eqn. 27. The numerical result for four eigenvalues of system I1 is listed in Table 7. Comparing with the contributions of the expectation and the variance, the higher order cumulant has slight effect on the eigen- value distribution, so the cumulant can be used as a hgher order numerical characteristic of an eigenvalue to largely reduce the computation requirement. From Table 7, a59 is unstable with probability 0.99865 under the normal assumption, but is stable with the correction of hgher order cumulants. For %5, the contribution of higher order cumulants to the conditional probability becomes as high as 3.427%. When the probability of an eigenvalue is close to unity, the effect of higher order cumulants becomes sig- nificant.

Table 7: Contribution of the 3rd. 4th and 5th order cumulants to conditional probability P{a c 0)

Ra,g < 0) < 01 P(a65 < 01 Ra74 < 01

From normal 0.99865 0.55962 0.92507 0.82121 distribution

Contributions of high 0.00135 0.00096 -0.03427 -0.00135 order cumulants

Final results 1 .OOOOO 0.56058 0.89080 0.81986

41

Because the computational requirement for higher order numerical characteristics of eigenvalues has been much reduced by cumulants, the proposed hybrid algorithm will be advantageous to the assessment of the stability degree for a multimachne system.

8 Conclusions

An improved probabilistic method for power system dynamic stability studies is proposed in this paper. The hybrid utilisation of central moments and cumulants ensures the consideration of both the dependence among input random variables and the correction for probabilistic densities of eigenvalues. According to the effects of various order cumulants on the distribution of an eigenvalue, dif- ferent degrees of approximation are applied for the calcula- tion of various order moments or cumulants. Second order terms are retained in the representation of eigenvalue expectations and used for the correction of variances, which improves the computing precision of the eigenvalue distribution and the corresponding conditional probability. The proposed method is examined on two test systems based upon operating curves of nodal powers. Comparison between the actual statistical nature of eigenvalues and the solution solved by the proposed method shows that ths method gives a satisfactory result and can be used for fur- ther eigenvalues studies.

9 Acknowledgments

The authors gratefully acknowledge the research funding provided by the Research Grants Council for ths project.

10 References

BURCHETT, R.C., and HEYDT, G.T.: ‘Probabilistic methods for power system dynamic stability studies’, IEEE Truns. Power Appur. Syst., 1978, PAS-97, (3), pp. 695702 BURCHETT, R.C., and HEYDT, G.T.: ‘A generalized method for stochastic analysis of the dynamic stability of electric power system’. IEEE PES, 1978, BRUCOLI, M., TORELLI, F., and TROVATO, M.: ‘Probabilistic approach for power system dynamic stability studies’, IEE Proc. C, 1981, 128, pp. 295-301 STANKOVIC, A.M., and LESIEUTRE, B.C.: ‘A probabilistic approach to aggregate induction machine modeling’, IEEE Trum. Power Syst., 1996, 11, (4), pp. 1983-1989

analysis’, J. Xim Jiuotong Univ., 1988, 22, (3) CHUNG, C.Y., WANG, K.W., CHEUNG, C.K., TSE, C.T., and DAVID, A.K.: ‘Machine and load modeling in large scale power industries. Dynamic modeling control applications for industry work- shop’. IEEE Industry Applications Society, April 1998, pp. 7-15 TSE, C.T., CHAN, K.L., HO, S.L., CHUNG, C.Y., CHOW, S.C., and LO, W.Y.: ‘Effective loadflow technique with non-constant MVA load for the Hong Kong mass transit railway urban lines power distri- bution system’. Proceedings of APSCOM-97, Hong Kong, 11-14 November 1997, pp. 753-757 “ D A L L , S.M. and STUART, A.: ‘The advanced theory of statis- tics, Volume 1’ (Hafner,’New York, 1977) LOU, G., XU, X., LONG, S., and LI, Q.: ‘Low frequency oscillations on the interconnectors between Hong Kong and Guangdong’, Proc. Chin. Soc. Electr. Enp., 1986. 6. (1). VP. 3Cb-35

WANG, X., and WANG, X.: ‘Power system probabilistic load flow

10 ZEIN EL-DIN, H.M., and ALDEN, R.T.H.: ‘Second order eigen- value sensitivities applied to power system dynamics’, IEEE Trans. Power Appar. Syst., 1977, PAS-%, (6), pp. 1928-1936

11 HIYAMA, T.: ‘Rule-based stabilizer for multimachine power sys- tem’, IEEE Trans. Power Syst., 1990, 5, (2), pp. 403-409

11 Appendices

11.1 State space equation Based on PMT [6], only two types of elementary transfer blocks as shown in Fig. 7 were contained.

Fig. 7

42

Two types of elementary t r m - r blocks

With vectors X,, X and M collecting all the xi, x and m variables of Fig. 7, R and W being the input and output vectors, the following connection matrix is constructed. [a] = [”: 2 21 [i] (19)

L7 L8 L9 The constructions of sub-matrixes L1 to L, are described

in detailed in [6], and these connection sub-matrixes are affected by input random factors. With a slight modifica- tion to the construction of L,, L8 and L9, the nodal admit- tance matrix Y describing the entire power network is directly inserted in the sub-matrix L9 with load powers being transferred to equivalent admittances. The state space equation is then formed, in which the system matrix A used for solving eigenvalues is determined from following equa- tions:

A = S ( K , F - Kb) (20)

s = ( I - K J - 1 (21)

F = Ll + L3(-Lg)-lL7 1122)

where K,, Kb and Kt are diagonal matrixes composed of the transfer block parameters kdTb, kdTb and T/Tb, respectively.

I 1.2 Probabilistic load flow As the nodal voltage vector V is defined in rectangular coordinates, the nodal injected power vector S(V) in a power system of N network nodes is a quadratic function of the nodal voltages. Accurately expanding S( V ) in a Tay- lor series, the second %der terms will have the same form as S(V). Considering AV = 0, the expected value will be

_ - S = S(v)+S(Av) = g ( . . . , V,V, +Cvt,, , . . .) (23)

where C , , is the covariance between voltages V, andl y. Therefore, the correcting equations used in the probabilistic load flow calculation are given as I

A S = JVAV ( 24)

(25) cv = J ; ~ c ~ ( J , -1 ) T

where J y is the Jacobian matrix. An alternative iteration of eqns. 24 and 25 takes account of the interaction between expectations and covariances.

11.3 Gram-Charlier series [81 When every order cumulants of a random variable have been determined, the probabilistic density and conditional probabilistic can be calculated from a Gram-Charlier series as eqns. 26 and 27, respectively.

K(4) K(5) +-(Z4 - 65’ + 3) + - (z5 - 10z3 + 152)

24a4 12005

P{” < zc} = I N ( z ) d z - N ( & ) [ - ;; (2: - 1) -cc

IEE Proc-Gener. Transm. Disfrib., Vol. 147, No. 1, January 2000

where o is the standard deviation of random variable x, R is the standardised x determined by

B = (X - T ) / g Zc = (zc - Z)/g (28) and N(.) stands for the density function of the standard normal distribution:

11.4 Partial derivatives of system matrix A From eqns. 2&22 the following matrix representation is obtained.

with a~ a - = - (L1+ L3HL7) av, av,

dL9 dL7 = LsH-HL7 + L3H- (32) av, av,

where H = (&-I.

In eqn. 31, the partial derivative of sub-matrix L, with respect to nodal voltages V, is composed of the derivatives

of sub-matrix elements, and so is that for sub-matrix L9 and the second order derivatives in eqn. 33. For a system of m generators, these sub-matrix elements associated with nodal voltages can be collected in a vector Z defined as

= [z(l), 2 ( 2 ) , . . . > z(m), Z G , ZBIT (34) Under the thud order machine model, Z(o corresponding to the ith generator is,

T z(t) [ E Q ~ , C O S ~ Z , S ~ ~ S Z , I ~ ~ , I ~ ~ ~ v d z , v q z , &,]

(35 ) The nodal powers are transferred to equivalent admittances in the connection (eqn. 19), which is a function of nodal voltages. ZG and Z, in eqn. 34 are just vectors collecting the real parts and imaginary parts of these equivalent admittances, respectively. The considerations of ZG and Z, were not very dflicult, but are neglected in some papers

Vector Z contains random factors for studying system dynamics. More random factors can be easily added to vec- tor Z if desired. In this paper, only the elements associated with nodal voltages are collected in vector Z, and the first and second order partial derivatives of these elements are derived from the following equations:

( equation Network) (36)

[I, 21.

I X = GVx - BVy I y = BVx + GVy

EQ, = V, + I , R, - I,X, EQ, = V, + I z X q +I&, E; = E$, + E;,

(Machine) (37) equation

COS S = EQ%/EQ sin6 = EQ,/EQ

I , = I, cos S + I , sin S

) (39) Coordinate

transformation

- I d = I , sin6 - I , cos6 V, =V,cosS+V,sinh Vd = V, sin6 - V,cosS vt” = v,” + vy”

) (40) Load model 2 = 1 , 2 , . . *

AY,, = AGii + jABii AYi, = - I i /v ,

where 6 is the power angle measured with respect to the slack busbar. Subscripts ‘q’ and ‘d refer to the machine d-q frame, whde ‘x’ and ‘y’ refer to the network frame.

IEE Proc-Gener. Transm. Distrib., Vol. 147, No. 1. January 2000 43